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2
(S3). The conditions (R3) and (S2) hold and det Ļā— (t) > 0 a.e. for
a.e. t.

We obtain theorems analogous to the preceding ones. In particular, if
a ā¤ b, a ā I, b ā I, then for an (S1) process
b
E{x(b) ā’ x(a) | Fb } = E Dā— x(s) ds Fb , (11.11)
a
KINEMATICS OF STOCHASTIC MOTION 79

and for an (S2) process
b
E{[yā— (b) ā’ yā— (a)] | Fb } = E Ļā— (s) ds Fb .
2 2
(11.12)
a

THEOREM 11.10 Let x be an (S1) process. Then

EDx(t) = EDā— x(t) (11.13)

for all t in I. Let x be an (S2) process. Then

EĻ 2 (t) = EĻā— (t)
2
(11.14)

for all t in I.

Proof. By Theorem 11.1 and (11.11), if we take absolute expectations
we ļ¬nd
b b
E[x(b) ā’ x(a)] = E Dx(s) ds = E Dā— x(s) ds
a a

for all a and b in I. Since s ā’ Dx(s) and s ā’ Dā— x(s) are continuous
in L 1 , (11.13) holds. Similarly, (11.14) follows from Theorem 11.4 and
(11.12). QED.

THEOREM 11.11 Let x be an (S1) process. Then x is a constant (i.e.,
x(t) is the same random variable for all t) if and only if Dx = Dā— x = 0.

Proof. The only if part of the theorem is trivial. Suppose that Dx =
Dā— x = 0. By Theorem 11.2, x is a martingale and a martingale with the
direction of time reversed. Let t1 = t2 , x1 = x(t1 ), x2 = x(t2 ). Then x1
and x2 are in L 1 and E{x1 |x2 } = x2 , E{x2 |x1 } = x1 . We wish to show
that x1 = x2 (a.e., of course).
If x1 and x2 are in L 2 (as they are if x is an (S2) process) there is a
trivial proof, as follows. We have

E{(x2 ā’ x1 )2 | x1 } = E{x2 ā’ 2x2 x1 + x2 | x1 } = E{x2 | x1 } ā’ x2 ,
2 1 2 1

so that if we take absolute expectations we ļ¬nd

E(x2 ā’ x1 )2 = Ex2 ā’ Ex2 .
2 1
80 CHAPTER 11

The same result holds with x1 and x2 interchanged. Thus E(x2 ā’x1 )2 = 0,
x2 = x1 a.e.
G. A. Hunt showed me the following proof for the general case (x1 , x2
in L 1 ).
Let Āµ be the distribution of x1 , x2 in the plane. We can take x1 and
x2 to be the coordinate functions. Then there is a conditional probability
distribution p(x1 , Ā·) such that if Ī½ is the distribution of x1 and f is a
positive Baire function on Ā‚2 ,

f (x1 , x2 ) dĀµ(x1 , x2 ) = f (x1 , x2 ) p(x1 , x2 ) dĪ½(x1 ).

(See Doob [15, Ā§6, pp. 26ā“34].) Then

E{Ļ•(x2 ) | x1 } = Ļ•(x2 ) p(x1 , dx2 ) a.e. [Ī½]

provided Ļ•(x2 ) is in L 1 . Take Ļ• to be strictly convex with |Ļ•(Ī¾)| ā¤ |Ī¾|
for all real Ī¾ (so that Ļ•(x2 ) is in L 1 ). Then, for each x1 , since Ļ• is strictly
convex, Jensenā™s inequality gives

Ļ• x2 p(x1 , dx2 ) < Ļ•(x2 ) p(x1 , dx2 )

Ļ•(x2 ) p(x1 , dx2 ) a.e. [p(x1 , Ā·)]. But
unless Ļ•(x1 ) =

x2 p(x1 , dx2 ) = x1 a.e. [Ī½],

so, unless x2 = x1 a.e. [Ī½],

Ļ•(x1 ) < Ļ•(x2 ) p(x1 , dx2 ).

If we take absolute expectations, we ļ¬nd EĻ•(x1 ) < EĻ•(x2 ) unless x2 = x1
a.e. The same argument gives the reverse inequality, so x2 = x1 a.e.
QED.

THEOREM 11.12 Let x be and y be (S1) processes with respect to the
same families of Ļ-algebras Pt and Ft , and suppose that x(t), y(t), Dx(t),
Dy(t), Dā— x(t), and Dā— y(t) all lie in L 2 and are continuous functions of
t in L 2 . Then
d
Ex(t)y(t) = EDx(t) Ā· y(t) + Ex(t)Dā— y(t).
dt
KINEMATICS OF STOCHASTIC MOTION 81

Proof. We need to show for a and b in I, that
b
E [x(b)y(b) ā’ x(a)y(a)] = E [Dx(t) Ā· y(t) + x(t)Dā— y(t)]dt.
a

(Notice that the integrand is continuous.) Divide [a, b] into n equal parts:
tj = a + j(b ā’ a)/n for j = 0, . . . , n. Then
nā’1
E [x(b)y(b) ā’ x(a)y(a)] = lim E [x(tj+1 )y(tj ) ā’ x(tj )y(tjā’1 )] =
nā’ā
j=1
nā’1
y(tj ) + y(tjā’1 )
E x(tj+1 ) ā’ x(tj )
lim +
2
nā’ā
j=1

x(tj+1 ) + x(tj )
y(tj ) ā’ y(tjā’1 ) =
2
nā’1
bā’a
E [Dx(tj ) Ā· y(tj ) + x(tj )Dā— y(tj )]
lim =
n
nā’ā
j=1
b
E [Dx(t) Ā· y(t) + x(t)Dā— y(t)] dt.
a

QED.

Now let us assume that the past Pt and the future Ft are condi-
tionally independent given the present Pt ā© Ft . That is, if f is any
Ft -measurable function in L 1 then E{f | Pt } = E{f | Pt ā© Ft }, and if f
is any Pt -measurable function in L 1 then E{f | Ft } = E{f | Pt ā© Ft }.
If x is a Markov process and Pt is generated by the x(s) with s ā¤ t, and
Ft by the x(s) with s ā„ t, this is certainly the case. However, the as-
sumption is much weaker. It applies, for example, to the position x(t) of
the Ornstein-Uhlenbeck process. The reason is that the present Pt ā© Ft
may not be generated by x(t); for example, in the Ornstein-Uhlenbeck
case v(t) = dx(t)/dt is also Pt ā© Ft -measurable.
With the above assumption on the Pt and Ft , if x is an (S1) process
then Dx(t) and Dā— x(t) are Pt ā©Ft -measurable, and we can form DDā— x(t)
and Dā— Dx(t) if they exist. Assuming they exist, we deļ¬ne
1 1
a(t) = DDā— x(t) + Dā— Dx(t) (11.15)
2 2
82 CHAPTER 11

and call it the mean second derivative or mean acceleration.
If x is a suļ¬ciently smooth function of t then a(t) = d2 x(t)/dt2 . This
is also true of other possible candidates for the title of mean acceleration,
such as DDā— x(t), Dā— Dx(t), DDx(t), Dā— Dā— x(t), and 1 DDx(t) + 1 Dā— Dā— x(t).
2 2
Of these the ļ¬rst four distinguish between the two choices of direction for
the time axis, and so can be discarded. To discuss the ļ¬fth possibility,
consider the Gaussian Markov process x(t) satisfying

dx(t) = ā’Ļx(t) dt + dw(t),

where w is a Wiener process, in equilibrium (that is, with the invariant
Gaussian measure as initial measure). Then

Dx(t) = ā’Ļx(t),
Dā— x(t) = Ļx(t),
a(t) = ā’Ļ 2 x(t),

but
1 1
DDx(t) + Dā— Dā— x(t) = Ļ 2 x(t).
2 2
This process is familiar to us: it is the position in the Smoluchowski de-
scription of the highly overdamped harmonic oscillator (or the velocity
of a free particle in the Ornstein-Uhlenbeck theory). The characteristic
feature of this process is its constant tendency to go towards the origin,
no matter which direction of time is taken. Our deļ¬nition of mean ac-
celeration, which gives a(t) = ā’Ļ 2 x(t), is kinematically the appropriate
deļ¬nition.

Reference

The stochastic integral was invented by ItĖ: o
[27]. Kiyosi ItĖ, āOn Stochastic Diļ¬erential Equationsā, Memoirs of the
o
American Mathematical Society, Number 4 (1951).
Doob gave a treatment based on martingales [15, Ā§6, pp. 436ā“451].
Our discussion of stochastic integrals, as well as most of the other material
of this section, is based on Doobā™s book.
Chapter 12

Dynamics of stochastic motion

The fundamental law of non-relativistic dynamics is Newtonā™s law
F = ma: the force on a particle is the product of the particleā™s mass
and the acceleration of the particle. This law is, of course, nothing but
the deļ¬nition of force. Most deļ¬nitions are trivialā”others are profound.
Feynman [28] has analyzed the characteristics that make Newtonā™s deļ¬-
nition profound:
āIt implies that if we study the mass times the acceleration and call
the product the force, i.e., if we study the characteristics of force as a
program of interest, then we shall ļ¬nd that forces have some simplicity;
the law is a good program for analyzing nature, it is a suggestion that
the forces will be simple.ā
Now suppose that x is a stochastic process representing the motion
of a particle of mass m. Leaving unanalyzed the dynamical mechanism
causing the random ļ¬‚uctuations, we can ask how to express the fact that
there is an external force F acting on the particle. We do this simply by
setting
F = ma
where a is the mean acceleration (Chapter 11).
For example, suppose that x is the position in the Ornstein-Uhlenbeck
theory of Brownian motion, and suppose that the external force is F =
ā’ grad V where exp(ā’V D/mĪ²) is integrable. In equilibrium, the particle
has probability density a normalization constant times exp(ā’V D/mĪ²)
and satisļ¬es
dx(t) = v(t)dt
dv(t) = ā’Ī²v(t)dt + K x(t) dt + dB(t),

83
84 CHAPTER 12

where K = F/m = ā’ grad V /m, and B has variance parameter 2Ī² 2 D.
Then

Dx(t) = Dā— x(t) = v(t),
Dv(t) = ā’Ī²v(t) + K x(t) ,
Dā— v(t) = Ī²v(t) + K x(t) ,
a(t) = K x(t) .

Therefore the law F = ma holds.

Reference

[28]. Richard P. Feynman, Robert B. Leighton, and Matthew Sands, āThe
1963.
Chapter 13

Kinematics of Markovian
motion

At this point I shall cease making regularity assumptions explicit.
Whenever we take the derivative of a function, the function is assumed
to be diļ¬erentiable. Whenever we take D of a stochastic process, it is
assumed to exist. Whenever we consider the probability density of a ran-
dom variable, it is assumed to exist. I do this not out of laziness but out
of ignorance. The problem of ļ¬nding convenient regularity assumptions
for this discussion and later applications of it (Chapter 15) is a non-trivial
problem.
Consider a Markov process x on Ā‚ of the form
dx(t) = b x(t), t)dt + dw(t),

where w is a Wiener process on Ā‚ with diļ¬usion coeļ¬cient Ī½ (we write
Ī½ instead of D to avoid confusion with mean forward derivatives). Here
b is a ļ¬xed smooth function on Ā‚ +1 . The w(t) ā’ w(s) are independent of
the x(r) whenever r ā¤ s and r ā¤ t, so that
Dx(t) = b x(t), t .
A Markov process with time reversed is again a Markov process (see
Doob [15, Ā§6, p. 83]), so we can deļ¬ne bā— by
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